New Aspects of the Bipolar Expansion and Molecular Multicenter Integrals

Standard bipolar expansions contain radial functions $J_{l_1{l}_{2}L}{\mathrm{(r}}_1{\mathrm{,\; r}}_2\mathrm{,\; R)}$ which depend on three variables, $r_1{\mathrm{,\; r}}_2\mathrm{,\; R}$ . Using representations of Bessel functions as series of Laguerre polynomials, the J's are expanded in terms of functions of the individual variables. As a result, the inverse distance between two points is brought into the form $$r_{12}^{\text{−1}}\mathrm{\hspace{0.17em}=\hspace{0.17em}}{\displaystyle \sum _{n_l{l}_l{m}_l}}{\displaystyle \sum _{n_2{l}_2{m}_2}}{F}_{n_1{l}_1{m}_1{\mathrm{,\; n}}_2{l}_2{m}_2}{\mathrm{(R,\; \theta ,\; \Psi )\; Y}}_{l_1{m}_1}{\mathrm{(\theta }}_1{\mathrm{,\; \phi }}_1{\mathrm{)f}}_{n_1{l}_1}{\mathrm{(r}}_1{\mathrm{)Y}}_{l_2{m}_2}{\mathrm{(\theta }}_2{\mathrm{,\; \theta }}_2{\mathrm{)f}}_{n_2{l}_2}{\mathrm{(r}}_2\mathrm{).}$$ This modified bipolar expansion is used to derive expressions for multicenter electron repulsion and nuclear attraction integrals. The method is particularly suitable for Gaussian orbitals expressed with spherical harmonics and yields compact expressions directly. For Slater‐type orbitals, the multicenter energy integrals appear as series involving only integrals of the two‐center overlap type. The one‐center and multipole limits of the bipolar expansion are examined.